22.2

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
22.2#Ex1	${\displaystyle{\displaystyle k=\frac{{\theta_{2}^{2}}\left(0,q\right)}{{\theta_{3}^{2}}\left(0,q\right)}}}$ k = \frac{\Jacobithetaq{2}^{2}@{0}{q}}{\Jacobithetaq{3}^{2}@{0}{q}}		k = ((JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2))/((JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2))	k == Divide[(EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2),(EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2)]	Failure	Failure	Error	Successful [Tested: 3]
22.2#Ex2	${\displaystyle{\displaystyle k^{\prime}=\frac{{\theta_{4}^{2}}\left(0,q\right)}{{\theta_{3}^{2}}\left(0,q\right)}}}$ k^{\prime} = \frac{\Jacobithetaq{4}^{2}@{0}{q}}{\Jacobithetaq{3}^{2}@{0}{q}}		sqrt(1 - (k)^(2)) = ((JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2))/((JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2))	Sqrt[1 - (k)^(2)] == Divide[(EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2),(EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2)]	Failure	Failure	Error	Successful [Tested: 3]
22.2#Ex3	${\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2}{\theta_{3}^{2}}\left(0,q\right)}}$ \compellintKk@{k} = \frac{\pi}{2}\Jacobithetaq{3}^{2}@{0}{q}		EllipticK(k) = (Pi)/(2)*(JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2)	EllipticK[(k)^2] == Divide[Pi,2]*(EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2)	Failure	Failure	Error	Failed [1 / 3] Result: DirectedInfinity[] Test Values: {Rule[k, 1]}
22.2.E3	${\displaystyle{\displaystyle\zeta=\frac{\pi z}{2K\left(k\right)}}}$ \zeta = \frac{\pi z}{2\compellintKk@{k}}		zeta = (Pi*z)/(2*EllipticK(k))	\[Zeta] == Divide[Pi*z,2*EllipticK[(k)^2]]	Failure	Failure	Error	Failed [210 / 210] Result: Complex[0.8660254037844387, 0.49999999999999994] Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.7059984047169785, -0.6365247818792681] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.2.E4	${\displaystyle{\displaystyle\operatorname{sn}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{2}\left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q\right)}}}$ \Jacobiellsnk@{z}{k} = \frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}}		JacobiSN(z, k) = (JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta1(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))	JacobiSN[z, (k)^2] == Divide[EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[1, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]	Failure	Failure	Error	Failed [140 / 210] Result: Complex[0.1017958925630662, 9.78035129055685*^-4] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.08293092681074243, -0.5359189266558633] Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.2.E4	${\displaystyle{\displaystyle\frac{\theta_{3}\left(0,q\right)}{\theta_{2}\left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q\right)}=\frac{1}{\operatorname{ns}\left(z,k\right)}}}$ \frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}} = \frac{1}{\Jacobiellnsk@{z}{k}}		(JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta1(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k)))) = (1)/(JacobiNS(z, k))	Divide[EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[1, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]] == Divide[1,JacobiNS[z, (k)^2]]	Failure	Failure	Error	Failed [140 / 210] Result: Complex[-0.1017958925630662, -9.780351290556814*^-4] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.08293092681074243, 0.5359189266558634] Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.2.E5	${\displaystyle{\displaystyle\operatorname{cn}\left(z,k\right)=\frac{\theta_{4}\left(0,q\right)}{\theta_{2}\left(0,q\right)}\frac{\theta_{2}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q\right)}}}$ \Jacobiellcnk@{z}{k} = \frac{\Jacobithetaq{4}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{2}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}}		JacobiCN(z, k) = (JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta2(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))	JacobiCN[z, (k)^2] == Divide[EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[2, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]	Failure	Failure	Error	Failed [140 / 210] Result: Complex[-0.08257811120249814, 0.0027270134984790223] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.13231049687767538, 0.2777560839806882] Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.2.E5	${\displaystyle{\displaystyle\frac{\theta_{4}\left(0,q\right)}{\theta_{2}\left(0,q\right)}\frac{\theta_{2}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q\right)}=\frac{1}{\operatorname{nc}\left(z,k\right)}}}$ \frac{\Jacobithetaq{4}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{2}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}} = \frac{1}{\Jacobiellnck@{z}{k}}		(JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta2(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k)))) = (1)/(JacobiNC(z, k))	Divide[EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[2, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]] == Divide[1,JacobiNC[z, (k)^2]]	Failure	Failure	Error	Failed [140 / 210] Result: Complex[0.08257811120249814, -0.002727013498479031] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.13231049687767538, -0.2777560839806882] Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.2.E6	${\displaystyle{\displaystyle\operatorname{dn}\left(z,k\right)=\frac{\theta_{4}\left(0,q\right)}{\theta_{3}\left(0,q\right)}\frac{\theta_{3}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q\right)}}}$ \Jacobielldnk@{z}{k} = \frac{\Jacobithetaq{4}@{0}{q}}{\Jacobithetaq{3}@{0}{q}}\frac{\Jacobithetaq{3}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}}		JacobiDN(z, k) = (JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta3(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))	JacobiDN[z, (k)^2] == Divide[EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[3, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]	Failure	Failure	Error	Failed [140 / 210] Result: Complex[-0.11217526698173597, -1.5044574583405517] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.6119897435833945, -2.3508894631681736] Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.2.E6	${\displaystyle{\displaystyle\frac{\theta_{4}\left(0,q\right)}{\theta_{3}\left(0,q\right)}\frac{\theta_{3}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q\right)}=\frac{1}{\operatorname{nd}\left(z,k\right)}}}$ \frac{\Jacobithetaq{4}@{0}{q}}{\Jacobithetaq{3}@{0}{q}}\frac{\Jacobithetaq{3}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}} = \frac{1}{\Jacobiellndk@{z}{k}}		(JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta3(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k)))) = (1)/(JacobiND(z, k))	Divide[EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[3, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]] == Divide[1,JacobiND[z, (k)^2]]	Failure	Failure	Error	Failed [140 / 210] Result: Complex[0.112175266981736, 1.5044574583405517] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.6119897435833943, 2.350889463168173] Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.2.E7	${\displaystyle{\displaystyle\operatorname{sd}\left(z,k\right)=\frac{{\theta_{3}^{2}}\left(0,q\right)}{\theta_{2}\left(0,q\right)\theta_{4}\left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{3}\left(\zeta,q\right)}}}$ \Jacobiellsdk@{z}{k} = \frac{\Jacobithetaq{3}^{2}@{0}{q}}{\Jacobithetaq{2}@{0}{q}\Jacobithetaq{4}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{3}@{\zeta}{q}}		JacobiSD(z, k) = ((JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k)))*JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta1(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta3(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))	JacobiSD[z, (k)^2] == Divide[(EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2),EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]*EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[1, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[3, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]	Failure	Failure	Error	Failed [138 / 210] Result: Complex[-0.10264566281694597, 1.7190366283522571] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.005017214212665183, 0.8218706074973681] Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.2.E7	${\displaystyle{\displaystyle\frac{{\theta_{3}^{2}}\left(0,q\right)}{\theta_{2}\left(0,q\right)\theta_{4}\left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{3}\left(\zeta,q\right)}=\frac{1}{\operatorname{ds}\left(z,k\right)}}}$ \frac{\Jacobithetaq{3}^{2}@{0}{q}}{\Jacobithetaq{2}@{0}{q}\Jacobithetaq{4}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{3}@{\zeta}{q}} = \frac{1}{\Jacobielldsk@{z}{k}}		((JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k)))*JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta1(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta3(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k)))) = (1)/(JacobiDS(z, k))	Divide[(EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2),EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]*EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[1, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[3, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]] == Divide[1,JacobiDS[z, (k)^2]]	Failure	Failure	Error	Failed [138 / 210] Result: Complex[0.1026456628169461, -1.7190366283522573] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.005017214212665148, -0.8218706074973681] Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.2.E8	${\displaystyle{\displaystyle\operatorname{cd}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{2}\left(0,q\right)}\frac{\theta_{2}\left(\zeta,q\right)}{\theta_{3}\left(\zeta,q\right)}}}$ \Jacobiellcdk@{z}{k} = \frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{2}@{\zeta}{q}}{\Jacobithetaq{3}@{\zeta}{q}}		JacobiCD(z, k) = (JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta2(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta3(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))	JacobiCD[z, (k)^2] == Divide[EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[2, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[3, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]	Failure	Failure	Error	Failed [140 / 210] Result: Complex[-0.23207264303523145, 2.174081147069575] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.3131092646447684, 1.178043032175558] Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.2.E8	${\displaystyle{\displaystyle\frac{\theta_{3}\left(0,q\right)}{\theta_{2}\left(0,q\right)}\frac{\theta_{2}\left(\zeta,q\right)}{\theta_{3}\left(\zeta,q\right)}=\frac{1}{\operatorname{dc}\left(z,k\right)}}}$ \frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{2}@{\zeta}{q}}{\Jacobithetaq{3}@{\zeta}{q}} = \frac{1}{\Jacobielldck@{z}{k}}		(JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta2(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta3(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k)))) = (1)/(JacobiDC(z, k))	Divide[EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[2, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[3, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]] == Divide[1,JacobiDC[z, (k)^2]]	Failure	Failure	Error	Failed [140 / 210] Result: Complex[0.23207264303523142, -2.174081147069575] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.3131092646447683, -1.178043032175558] Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.2.E9	${\displaystyle{\displaystyle\operatorname{sc}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{4}\left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{2}\left(\zeta,q\right)}}}$ \Jacobiellsck@{z}{k} = \frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{4}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{2}@{\zeta}{q}}		JacobiSC(z, k) = (JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta1(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))	JacobiSC[z, (k)^2] == Divide[EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[1, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]	Failure	Failure	Error	Failed [140 / 210] Result: Complex[0.2180891710993932, -0.009050644683206828] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.13880139985550538, -0.6261898650931494] Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.2.E9	${\displaystyle{\displaystyle\frac{\theta_{3}\left(0,q\right)}{\theta_{4}\left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{2}\left(\zeta,q\right)}=\frac{1}{\operatorname{cs}\left(z,k\right)}}}$ \frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{4}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{2}@{\zeta}{q}} = \frac{1}{\Jacobiellcsk@{z}{k}}		(JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta1(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k)))) = (1)/(JacobiCS(z, k))	Divide[EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[1, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]] == Divide[1,JacobiCS[z, (k)^2]]	Failure	Failure	Error	Failed [140 / 210] Result: Complex[-0.2180891710993933, 0.009050644683206842] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.13880139985550533, 0.6261898650931494] Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.2.E11	${\displaystyle{\displaystyle\operatorname{pq}\left(z,k\right)=\ifrac{\theta_{p}\left(z\middle|\tau\right)}{\theta_{q}\left(z\middle|\tau\right)}}}$ \genJacobiellk{p}{q}@{z}{k} = \ifrac{\Jacobithetatau{p}@{z}{\tau}}{\Jacobithetatau{q}@{z}{\tau}}		genJacobiellk(p)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))* z*k = (JacobiThetap(z,exp(I*Pi*tau)))/(JacobiThetaexp(- Pi*EllipticCK(k)/EllipticK(k))(z,exp(I*Pi*tau)))	genJacobiellk[p]*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])* z*k == Divide[EllipticTheta[p, z, Exp[I*Pi*(\[Tau])]],EllipticTheta[Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]], z, Exp[I*Pi*(\[Tau])]]]	Failure	Failure	Error	Failed [300 / 300] Result: Plus[Times[Complex[0.5000000000000001, 0.8660254037844386], genJacobiellk], Times[Complex[-0.31964140165035193, 0.682988488811487], EllipticTheta[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Complex[0.8660254037844387, 0.49999999999999994], Complex[-0.1897367196265697, 0.08493465422971205]]]] Test Values: {Rule[k, 1], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Times[Complex[0.26976733074627424, -0.3419272748333145], genJacobiellk], Times[-1.0, EllipticTheta[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Complex[0.8660254037844387, 0.49999999999999994], Complex[-0.1897367196265697, 0.08493465422971205]], Power[EllipticTheta[Power[E, Times[-1, Pi, EllipticK[-3], Power[EllipticK[4], -1]]], Complex[0.8660254037844387, 0.49999999999999994], Complex[-0.1897367196265697, 0.08493465422971205]], -1]]] Test Values: {Rule[k, 2], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.2.E12	${\displaystyle{\displaystyle\tau=\ifrac{\mathrm{i}{K^{\prime}}\left(k\right)}{K\left(k\right)}}}$ \tau = \ifrac{\iunit\ccompellintKk@{k}}{\compellintKk@{k}}		tau = (I*EllipticCK(k))/(EllipticK(k))	\[Tau] == Divide[I*EllipticK[1-(k)^2],EllipticK[(k)^2]]	Failure	Failure	Error	Failed [30 / 30] Result: Complex[0.8660254037844387, 0.49999999999999994] Test Values: {Rule[k, 1], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.4867361401447923, 0.0147898206680519] Test Values: {Rule[k, 2], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data